MB March 10, 2002

1. MB March 10, 2002

2. MB March 10, 2002 Model Two requirements for a good model 1: It must accurately describe a large class of observations with few arbitrarily elements. 2: It must make definite predictions about the results of future observations. (extracted from Stephen Hawking ‘Á brief history of time,’ Bantam Books)

3. MB March 10, 2002 Types of Models (CE544) ELASTIC LINEAR NON-LINEAR PLASTICITY NON-LINEAR CAM-CLAY FAMILY Elasto-plastic

4. MB March 10, 2002 NON-LINEARITY GEOMETRIC – change of shape, size, etc. MATERIAL – change of properties CAUSES: stress state. History of loading, change in stiffness, physical conditions, in situ stress, water content, voids ratio

5. MB March 10, 2002 ELASTIC MODELS LINEAR – magnitude of response proportional to excitation Non-LINEAR – magnitude of response not proportional to excitation

6. MB March 10, 2002 CONSTITUTIVE LAWS SET OF EQUATIONS THAT RELATE STRESSES TO STRAINS. F(stress, stress rate, strain, strain rate) = 0 Homogeneity of time.

7. MB March 10, 2002 CONSTITUTIVE EQUATIONS

8. MB March 10, 2002 ELASTIC MODELS Elastic materials: State of stress is a function of the current state of deformation; no history effects Cauchy – stress is a function of strain (infinitesimal strain, first order) Green – based on strain energy function (Hyper-elastic)

9. MB March 10, 2002 Hooke’s law – simple case Simple one dimensional case: E = Young’s modulus (Elastic modulus)

10. MB March 10, 2002 Hooke’s law – General state

11. MB March 10, 2002 Shear stresses and strains

12. MB March 10, 2002 Typical values of E and G

13. MB March 10, 2002 Principal stresses

14. MB March 10, 2002 AXISMMETRIC CONDITION

15. MB March 10, 2002 PLANE STRAIN CONDITION

16. MB March 10, 2002 Hooke’s law using stress invariants

17. MB March 10, 2002 Constitutive elastic model – stress invariants

18. MB March 10, 2002 Lame’s constant

19. MB March 10, 2002 Poisson’s ratio

20. MB March 10, 2002 Green’s elastic model The work done by external forces in altering the configuration of a body from its natural state is equal to the sum of the kinetic energy and the strain energy

21. MB March 10, 2002 ANISOTROPIC ELASTICITY Anisotropic materials have different elastic parameters in different directions. Structural anisotropy or transverse anisotropy – manner in which soil is deposited. Stress induced anisotropy – differences in normal stresses in different directions.

22. MB March 10, 2002 Transverse anisotropy- most prevalent in soils

23. MB March 10, 2002 ELASTICITY AND PLASTICITY Theory of elasticity: uniqueness – behavior of the material expressed by a set of equations Theory of plasticity: discontinuity in stress-strain relationship (involves discontinuities and inequalities); deals with initial stress problems, state of structure at collapse, at post-yield.

24. MB March 10, 2002 THEORY OF PLASTICITY TO ADEQUATELY DESCRIBE THE PLASTIC DEFORMATION OF SOILS TO USE RELATIONSHIPS DEVELOPED TO PREDICT FAILURE LOADS AND SETTLEMENT.

25. MB March 10, 2002 PLASTIC RESPONSES

26. MB March 10, 2002 FULL PLASTIC STATE (COLLAPSE) Guess a plastic collapse mechanism For small deformation of this mechanism, integrate the work consumed in plastic deformation over the whole body Equate this to the work supplied to find the collapse load (ref: Calladine, C. R. “Engineering plasticity”, Pergamon Press, London)

27. MB March 10, 2002 PLASTICITY THEOREMS LOWER BOUND –IF ANY STRESS DISTRIBUTION THROUGOUT THE STRUCTURE CAN BE FOUND WHICH IS EVERYWHERE IN EQUILIBRIUM INTERNALLY AND BALANCES CERTAIN EXTERNAL LOADS AND AT THE SAME TIME DOES NOT VIOLATE THE YIELD CONDITION, THESE LOADS WILL BE CARRIED SAFELY BY THE STRUCTURE. UPPER BOUND –IF AN ESTIMATE OF THE PLASTIC COLLAPSE LOAD OF A BODY IS MADE BY EQUATING INTERNAL RATE OF DISSIPATION OF ENERGY TO THE RATE AT WHICH EXTERNAL FORCES DO WORK IN ANY POSTULATED MECHANISM OF DEFORMATION OF THE BODY, THE ESTIMATE WILL BE EITHER HIGH, OR CORRECT.